NSF (to DRB and JPS), and by an NIH Postdoctoral Fellowship. (to HLB). parameters ... the elementary building block of the nervous system when viewed as a network of interacting ...... Center for the Health Sciences. University of California.
Biol. Cybernetics 22, 213 228 (1976) @ by Springer-Verlag 1976
Identification of Synaptic Interactions* David R. Brillinger Department of Statistics, University of California, Berkeley, California H u g h L. B r y a n t , Jr., a n d Jos6 P. S e g u n d o Department of Anatomy and the Brain Research Institute, University of California, Los Angeles, California USA Received: December 2, 1975
Abstract This paper studies the influence exerted by the presynaptic spike train on the postsynaptic one. It applies to synaptic exploration a novel method for characterization of point-process systems (Brillinger, 1974, 1975a), and draws from it physiologically meaningful conclusions. The departure point was a large data set of action potential trains from an Aplysia network whose neurons are connected by monosynaptic inhibitory or excitatory PSP's, and either discharged spontaneously or were driven by intracellular pulses. First, a sequence of "kernels" is estimated, each with a physiological connotation relevant to synaptic transmission. The kernel independent of time - of zero-order - measures the postsynaptic rate with no presynaptic discharge. That of a single time argument - of first-order - relates to the rate effect of the average PSP. Those of two, three, or more time arguments - of second, third or higherorder - relate to interactions between two, three, or more postsynaptic potentials (e.g. to facilitation) and/or spikes (e.g. to refractoriness). Then successive models are constructed recursively and based on the kernel of zero-order, on the kernels of zero and first order, on those of zero, first and second order, and so forth, until a desired approximation is achieved. The plausibilities of each kernel estimate and of each model are evaluated separately by way of spectra and coherences. The "linear" model based upon the zero and first-order kernel was tested (after that based exclusively on the zero-order one was proven inadequate). When presynaptic discharges are very irregular and at intermediate or low rates, it provides satisfactory description and prediction, and the first-order kernel is an uncontaminated display of the rate effects of the average presynaptic spike: this constitutes the "linear" domain. When presynaptic discharges are bursty, regular or very fast, the linear model is unsatisfyctory: this is referred to as "non-linear" domain. Reasons for non-linearity lie in PSP facilitation and anti-facilitation, conversion of membrane current into firing rate, after-spike excitability oscillations, and special pacemaker interactions. The model can be extended to three-neuron networks where partial coherences extract interactions between followers, even while submitted to a common driver. The basic and ubiquitous issues of spike train description and stability were discussed. The counting and the interval statistics of spike trains provide equivalent descriptions and their current opposition is conceptually meaningless. Concomitant short-term fluctuations in spike generation intensity at preand postsynaptic levels have functional significance beyond changes in the overall average rate or interval: they are made precise by * Supported by grants from NIH (to JPS), UCP (to JPS) and NSF (to DRB and JPS), and by an NIH Postdoctoral Fellowship (to HLB).
parameters whose definition, estimation and physiological interpretation are presented here. Some stability of the experimental preparation is presupposed by investigators, but variations (e.g. from cycles or deterioration) always exist. Hence, decisions as to the preparation's evolution and as to tolerable changes must be made, and based upon pre-existing knowledge, educated guesses and practical considerations. This study provided basic knowledge of the individual synapse considered the elementary building block of the nervous system when viewed as a network of interacting nerve cells. It also contributed generally applicable mathematical techniques which were illustrated by application to relatively well studied and simple networks.
Introduction The present communication describes how action p o t e n t i a l (AP) d a t a c a n be u s e d as indices of s y n a p t i c f u n c t i o n , a l l o w i n g t h e i n v e s t i g a t o r to d e s c r i b e t r a n s s y n a p t i c r e l a t i o n s a n d to m o d e l or " i d e n t i f y " s y n a p t i c o p e r a t i o n t h r o u g h a n e x p a n s i o n t h a t relates p h y s i o l o g ically m e a n i n g f u l variables. S y s t e m i d e n t i f i c a t i o n rests u p o n a vast q u a n t i t a t i v e m e t h o d o l o g y (e.g. A s t r o m a n d Eykhoff, 1971 ; N i e m a n et al., 1971), b u t m o s t of it refers to l i n e a r o p e r a t i o n s w h e r e the i n p u t a n d o u t p u t f u n c t i o n s are c o n t i n u o u s o r d e f i n e d at d i s c r e t e e q u i s p a c e d points. L i v i n g systems, h o w e v e r , often are n o n - l i n e a r a n d e x p r e s s themselves by point-like processes (Marmarelis and a n d N a k a , 1973a, b; S e g u n d o , 1971). T h e m e t h o d s u s e d h e r e for s y n a p t i c i d e n t i f i c a t i o n , b a s e d in p a r t u p o n p r o c e d u r e s d e v e l o p e d by B r i l l i n g e r (1974, 1975a), a r e d e s i g n e d specifically for p o i n t p r o c e s s s y s t e m s a n d m a k e n o a s s u m p t i o n as to w h e t h e r o r n o t t h e y are linear. R e l a t e d a p p r o a c h e s are d e s c r i b e d by P e r k e l et al. (1970), T e r z u o l o (1970), M a r m a r e l i s a n d N a k a (1973a), K n o x a n d P o p p e l e (1975, a n d K r a u s z (1975). T h e s y n a p s e is " i d e n t i f i e d " w h e n c e r t a i n r e l e v a n t f u n c t i o n s a r e d e r i v e d f r o m c o r r e s p o n d i n g pre- a n d p o s t s y n a p t i c trains. T h e a c c e p t a b i l i t y of e a c h m o d e l is m e a s u r e d by c o h e r e n c e functions. T h e first app r o x i m a t i o n c l a i m s t h a t the rate o r p r o b a b i l i t y of
214 the postsynaptic spike relates only to a function that reflects the postsynaptic rate change after each presynaptic arrival, and to the rate were there no input. Better approximations are achieved when complexities due to PSP interactions, to postsynaptic after-firing excitability shifts, etc. are oonsidered too. The model admits extension to multicell networks with, say, one driver cell and two followers, where partial coherences measure the degree of association of the followers with the driver's effects removed. A special section defines and explains the parameters used in these studies with AP trains, indicating their estimation procedures and physiological interpretations. Also, it discusses briefly the basic issues of spike train description and experimental stability. A preliminary report has appeared recently (Segundo et al., 1975).
Experimental Methodology The preparation (abdominal ganglion of Aplysia californica), experimental procedure, and electrophysiological recording and stimulation techniques were identical to those in a previous communication (Bryant et al., 1973). The present report re-examines much of the data used therein. In summary, the isolated abdominal ganglia of small specimens were perfused in a buffered Aplysia saline at a servocontrolled temperature between 14 and 21 ~ C. Intracellular recordings and passage of current were obtained with potassium citrate-filled electrodes, a bridge circuit and conventional amplification, display and analog tape storage devices. In a typical case, the presynaptic cell L 10 was impaled with one or two "followers" (e.g. L 3, L 2). In "spontaneous" experiments, activity was recorded with no deliberate stimulation (or, at most, with a slight DC accelerating or slowing bias). In "driven" experiments, the presynaptic cell was stimulated intracellularly with 50 msec duration rectangular pulses, each eliciting one AP. Each interspike interval was drawn independently from a prespecified density: different presynaptic discharge "forms" reflected different densities which were approximately exponential, narrow Gaussian or uniform and are referred to as "Poisson", "pacemaker", or "uniform", respectively. All densities had mean intervals between 0.1 and 2 sec (implying rates between 0.5 and 10 per sec), and were truncated at a lower bound of 50msec (because of refractoriness). The L 10 junctions with EPSP's (e.g. at R15 or R16) either are hard to find or are labile: hence, the relatively stable EPSP elicited by a single shock to right viscero-pleural connective (RVP) in R 15 was analyzed too.
A total of 47 cases were studied. Each case implied observation of two, or sometimes three corresponding and stationary AP trains, one presynaptic and one or two postsynaptic. The presynaptic discharge form was "Poisson" in 19 cases, "uniform" in 13, "pacemaker" in 8 and "bursty" in 7. To illustrate this communication we have deliberately chosen cases that were representative, i.e. that reflected the majority in their class, and were not necessarily the most striking (e.g. had the highest coherences, see below). Whenever possible, we deliberately chose data used for the figures of Bryant et al. (1973), considering this an economic way of comparing different methods.
Statistical Methodology Model Description; Basic Issues; Parameter Definition, Estimation and Physiological Interpretation The purpose of this communication is to provide a quantitative method for an exhaustive and physiologically meaningful characterization of the synapse as a system. Hence, the core of this section is the modelling and "identification" approach with the definition, estimation procedures and physiological implication of certain statistics, including methods not available elsewhere in complete form and laid down here in enough detail to be of use to physiologists. Two general issues are pertinent, spike train description and stability. They are, in fact, very basic to a large number of publications, imposing decisions which, deliberate or unknowingly, are implied always though rarely made explicit. Their relevancy and generality are so great that it was felt that they should not remain undiscussed. The first refers to the representation of AP trains. The empirical observation of the importance of the times of occurrence of AP's, commonly all-or-nothing, brief with respect to the interval between them, justifies for certain purposes the assimilation of a train to a point process along a line. Such a process is described fully by an ordered sequence of the times ... < a_ 2 < (7-1 < o'0 < (71 < (72 < ... of occurrence of each point (Cox and Lewis, 1966). Inherent and central in the intuitive notion of any such process is whether the points along the line are abundant and closely packed, or sparse and widely separated, i.e. the "intensity" of their generation. Also intuitive is the recognition that the intensity is linked directly with the likelihood of encountering a point. A first representation of a point process is through the "counting process" N(t), i.e. the number of points between time 0 and time t. As t increases, this variate jumps by one at each point. The overall intensity is reflected by the mean rate of the process, estimated by
215
for large T. A second representation of the point process, intuitively less natural, is provided by the "ordered sequence of intervals" ..., iol = ~1 - 00, i a2 = ~rz5-.cr~.... between successive points. The overall intensity is reflected reciprocally by the average interval estimated by T(n)/n, where n is the number of intervals observed and T(n) = Oo + io, 1 + ... + i,_ 1,, is the time from the origin to the last point, The mean rate itself may be estimated by niT(n). Both of these exhaustive representations are essentially equivalent (Cox and Lewis, 1966); however in practice, when simple properties are studied, both can be informative and the approaches complementary. Two conclusions can be drawn. First, that the opposition of counting versus interval statistics, as implied in for example Terzuolo and Bayly (1968), is not meaningful conceptually since it contraposes equivalent descriptions. Indeed, any question and answer can be expressed interchangeably in count or interval language. Second, that in a practical situation either counting or interval statistics may provide the more appropriate quantification and parsimonious description for the central notion of firing intensity. Neurophysiologists often have been satisfied with observing whether there were a lot or only a few spikes over a relatively prolonged period, using the overall mean rate as a measure; important concepts have been clarified through this approach. An experiment usually provides a number of trains, collected so as to appreciate whether changes in the discharge and in some sensory or motor variable can be matched appropriately. There usually are, however, within these relatively prolonged periods, shorter term variations in AP intensity. It is here that the really meaningful physiological question arises, asking whether the overall mean rate suffices to infer or predict the corresponding cause or consequence (e.g. sensory stimulus or movement), or the shorter term rate or interval fluctuations, i.e. the spike timing and pattern, are meaningful and informative too (e.g. Segundo and Perkel, 1969). The latter are revealed by a running display of rates over shorter periods, or of successive intervals, and can be summarized by, say, an autointensity function or an interval standard deviation. Experimental evidence supports the intuitive belief in the second alternative (e.g. Segundo et al., 1963; Segundo, 1970). Thus, a second important question arises: namely, that of how fluctuations are transferred in terms of amplitudes and time courses. Efforts to analyze these questions in the sensory sphere are abundant, since the early work of Pringle and Wilson (1952). The same questions can be asked a p r o p o s of synaptic transfer, and paragraphs below make precise N(T)/T
the notion of short term discharge intensity variations, concomitant in pre- and postsynaptic neurons, describing a method for identification of point process systems. The second issue, stability, refers to the extent to which preparations depart as time goes on from the characteristics exhibited initially. The investigator assumes, on the one hand, that the system realized by the preparation has invariant features that allow for general conclusions; the statistician formalizes this in a requirement of stationarity (Cox and Lewis, 1966). There are, however, inevitable instabilities in living matter; for example, aging and the deterioration of the experimental preparation. All systems Vary during the observation time by trends and/or fluctuations and each instance raises such questions as whether an observed change is small enough that the preparation can still be considered stable, whether a progressive change is part of a trend or of a cycle that eventually would have reverted to the initial conditions, and so forth. The unavoidable judgement concerning acceptable stability thus requires an a priori practical decision as to the magnitude and quality of tolerable changes (La Salle and Lefschetz, 1961; O'Leary et al., 1975; Weiss and Infante, 1967) and as to the expected form of the preparation's evolution. These decisions with their important connotations are reached for each case on the basis of pre-existing knowledge, educated guesses and practical considerations. The model arises from methods recently formalized and made practical (Brillinger, 1975a) for identification of point process systems, that is of systems whose input and output are point processes (the synapse and spike trains, respectively, in this case). The synapse is said to be "identified" when an acceptable model is found. One model involves an expansion based upon functions referred to as "kernels", and is a point process analog of the Volterra-Wiener expansion of ordinary time series (Marmarelis and Naka, 1973a, b). The kernels are i. expressed as functions of time arguments, ii. meant to be invariants of the system that retain the same essential characteristics even when the presynaptic discharge varies (other commonly used functions do not have this property, e.g. Bryant et al., 1973; Knox, 1974; Moore et aI., 1970), and iii. estimated from corresponding pre- and postsynaptic spike trains. The first step of the identification is to estimate certain conditional rate functions, each of which is a physiological connotation relevant to synaptic transmission. The one of zero-order g, i.e. a constant, simply measures the mean rate. The one of first-order,
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a function of a single time argument, relates to the average effect of a presynaptic spike and the PSP it elicits. The one of second-order, a function of two time arguments, relates to the interactions between pairs of spikes or PSP's. Those of higher orders, functions of several time arguments, relate to interactions between more than two events. The second step is to construct recursively the successive models, i.e. that based on the zero-order kernel, that based on the zero- and first-order, that based on the zero-, first- and second-order, and so on until it makes sense to add no more. The acceptance of a kernel estimate as plausible does not necessarily mean that the series up to the corresponding term is a good predictor. The coherence provides a measure of predictability. The kernel of first-order a(u) relates to the effect of a single presynaptic spike or PSP. It is defined as the best linear predictor of the average change of the instantaneous rate at time u in a spike train B, when a single spike occurs at time 0 in a spike train A. It is useful, therefore, in predicting whether there will be a B spike u time units away from an A spike, and will be positive, negative or zero, if B accelerates, slows or does not change, respectively, a(u) satisfies
mAB(U)-- mB= a(u) + ~ a(u-- v)[mAA(V) -- mA]dV
(1)
the expression prob {B spike in (t, t + h)[ A spikes at ~j} ~[/1+
~
a(t-@]h.
(3)
j = --oo
The constant g represents B's rate with A inactive. The rationale for (3) follows: if there were no A spikes, the probability of a B spike close to t would be gh, being B's rate with A silent; if the A train consisted of a single spike at time a, it would be [#+a(t-~)]h for some function a( .); if it consisted of two spikes at times al, a2, and they did not interact (see below), the probability would be [/~ + a(t - al) + a ( t - o-2)]h. Extending this reasoning to a train having many non-interacting spikes, the probability would be
Ilt+=~_ooa(t-~r,)lh
(4)
as h$0. The summation is extended from minus to plus infinity for reasons given below. Averaging expression (3) over all possible A trains leads directly to the first-order relationship
m B= # + m A ya(v)dv.
(5)
where - o o < u < o o . mAB(U) is the AB cross-intensity function (CIF), a first-order conditional rate function that measures, for one cell B and close to any particular time t (i.e. between t and t+h, with h positive and tending to zero), the average instantaneous rate or the likelihood of generating an AP, conditional on an A spike u time units away (e.g. Bryant et al., 1973; Knox, 1974; Moore et al., 1970). Its profile reflects the timing and the rate effects of A spikes, among other issues (as the correlation of A with a third cell C which also acts upon B). It is defined by
Multiplying Eq. (3) through by the differential increment dNa(t-u) of the counting process, averaging and using the identity (5) leads to the integral Eq. (1). A second derivation of (1) comes about from seeking to predict the instantaneous rate of the B train from the times of A spikes by an expression of the form (4), i.e. without interactions. If NB(t, t+h) denotes the number of B spikes in (t, t + h), we ask for the # and a(u) that lead to the smallest separation (measured by the average of the squared differences) between the instantaneous rate, NB(t, t+h)/h, and a postulated function of the form i~+Za(t-@: i.e. that lead to the minimum of
mAB(u) = lim prob {B spike in (t, t + h) I A
lim E [ N B(t, t + h)/h- # - Z a ( t - a j)12 . h$0
h$0
spike at t - u } / h ,
mAA(U)is the
(2)
auto-intensity function of A, i.e. the CIF of the A train with itself. The constants m A and m8 are the overall mean rates. The integral Eq. (1) can be derived and justified in two distinct manners, both relevant to synaptic characterization. Suppose that o-j(j=0, +_ 1, • .... ) denote the times of the presynaptic spikes and zk(k = O, +_1, +--2, ...) those of the postsynaptic spikes. In the first derivation, we set about modelling the change in the likelihood of a postsynaptic AP very close to t by
(6)
Evaluating expression (6) and using the calculus of variations leads again to the integral Eq. (1). The first-order kernel a(u) is, in all cases, the best linear predicter in the sense of expression (3), and one might anticipate that it would be zero for negative times when A acts trans-synaptically on B. It is necessary to understand, however, that the procedure under discussion is designed to fit associations and, therefore, no particular kernel will necessarily suffice to describe certain effects. Hence, though this may indeed happen, in other cases unexpected features may appear; for example, when studying pacemaker cells
217
there may be "predictive" features which seem to ignore causality like non-zero a(u) values to the left of the origin. This, and the fact that the direction of causation is sometimes unknown, is the reason why the summation in Eq. (4) is from - o o to or. Both derivations of a(u) used the assumption that the effects of separate A spikes are additive and not interactive in their influences upon B rates. This clearly holds when, for example, the presynaptic discharge is slow and successive spikes are so far apart that the effects of each have died down by the time the next arrives. Under these conditions, model (3) is plausible and the first-order kernel can be referred to as "average impulse response function" of the system. On the other hand (see Results), a(u) is subject to limitations that arise from not allowing for common and often powerfully interactive issues (Segundo et al., 1963) like the after-effects of earlier presynaptic arrivals, e.g. after-firing fluctuations of exitability. Hence, it may be necessary to extend the model to include higher-order terms. A conditional rate function of second-order allows in part for these issues: t T I A A B ( ' U , I)) ~ -
lim prob {B spike in (t, t + h) I A (7)
This function represents the postsynaptic rate close to time t, conditional on the presynaptic cell having fired u and v seconds earlier, lit may be estimated by expressions analogous to (13) below, and the large sample properties remain including the advantages of taking square roots, Brillinger, 1975b.] When, as in Eq. (7), two presynaptic spikes are taken into account, the probability of a postsynaptic spike close to t, given a presynaptic train of spikes at time cry,can be modelled by
[#+Sjaa(t--aj)+Fka2(t--aj, t--~rk)]h.
z +
2
ak(t-aj ...... t-aj,,)+ ...Ih
J l ~=j2 47 9 . 9 j k distinct
(9)
3
where the generic kernel a k incorporates the interaction at time t of the effects of k distinct presynaptic events at times as1,.., as,~. When k = 1, (9) reduces to (4), which is referred to as the "linear" model. Another set of conditional rate functions and kernels takes into account postsynaptic firings with their resulting excitability shifts (e.g. refractoriness). Yet another takes into account combinations of preand postsynaptic firings. For example, a useful "mixed" function of two time arguments is provided by:
mABB(U, V)= lim prob {B spike in (t, t + h) [ A h;O
h J, 0
spikes at t - u and t-v}/h.
of the synapse's behavior over a reasonable domain of inputs has been achieved. "Suitability" is evaluated by means of the coherence, a general measure of association applicable to models of any order (see below). The general model based upon presynaptic spikes (or PSP's) is of the form
(8)
a2(u, v) is the second-order kernel whose arguments are the times to distinct presynaptic spikes. It relates to the rate effects of two presynaptic spikes combined at any given relative timing, incorporating into the identification the influence that one presynaptic spike (say, at t - u ) exerts upon the rate effects of another (at t-v). It thus is sensitive to PSP facilitation and anti facilitation. A new term is added to (8) for each new presynaptic spike (or PSP) one wishes to account for. Thus, higher order kernels incorporate the consequences of PSP facilitation or anti-facilitation that occur after two, three, ... events with any conceivable given timing. The expression is expanded until a suitable description
spike at t - u, B at t - v}/h.
10)
This function represents the postsynaptic rate close to t, conditional on the presynaptic cell and the postsynaptic cell having fired u and v seconds earlier, respectively. It and the corresponding kernel are useful for identification when there exists an influence of a postsynaptic spike (at t - v ) upon the rate effects of a presynaptic one (at t - u ) ; or, equivalently, an effect of a single A firing (at t - u ) upon B's rate at two instants ( t - v , t). Thus, they are sensitive to the joint after-effects of postsynaptic potentials and spikes. The zero- and the first-order kernel of (1) are examined in this paper. A forthcoming publication shall explore higher orders (Brillinger, Bryant and Segundo, in preparation). Models of all orders require essentially the same methodology; computational problems may arise mainly because of the large number of operations and storage requirements needed in their evaluation. A requirement important for ease of interpretation is that each successive term in an expression for the probability not be affected greatly by factors accounted for by earlier terms. Other procedures (e.g. Bryant et al., 1973; Knox and Poppele, in preparation; Perkel, 1970; Perkel et al., in the press) allow at best a partial separation of issues. Models (3), (8) and (9) for the probability of a postsynaptic spike conditional on the presynaptic train are direct
218 expansions which, in essence, imply that the influences certain parameters. Suppose that A is a stationary of the various issues are combined by an additive spike train. Let Na(t ) denote the number of A spikes process. It is conceivable, however, that a more 1 between time 0 and time t. The mean intensity of the reasonable description could be obtained by other A train is defined by expansions implying combination by, for example, a prob {A spike in (t, t+ h)}/h. (11) multiplicative process (so that the logarithm of the m a = hlim ,~0 probability and not the probability itself is given by If the train is observed for the time interval 0 < t < T, a sum). This is an empirical question, and research is then m A may be estimated by in progress to see which expansion is most suitable (Brillinger, Bryant and Segundo, in preparation). rhA = NA(T)/T. (12) There are other related efforts in the literature approaching the same questions. For example, Perkel The cross-intensity function, mAB(U), between two (1970) proposed that, since pre- and postsynaptic simultaneously occurring spike trains A and B was correlations depend on the presynaptic rhythmicity defined above by expression (2). Because of stationarity and on the primary effects, the cross-covariance could this function does not depend on t. It gives the shortbe the convolution of the presynaptic auto-covariance term intensity of the B train u time units after an A and of a time function e(u) called the "synaptic re- spike, If the trains are independent, then mAB(U)= me for all u. IfB spikes are independent of"later" A spikes, sponse". The latter in turn depends on a term a(u) that reflects primary PSP effects, and on the resetting then mAB(U)=mB for all u
